Reza Nikandish

Let [ ] be a commutative ring with unity, [ ] be the set of all ideals of [ ] and [ ] be a subset of [ ]. Cayley sum graph of ideals of [ ], denoted by [ ], is a simple undirected graph with vertex set [ ] and two distinct vertices [ ] and [ ] are adjacent ([ ]), whenever [ ] or [ ], for some ideal [ ] in [ ]. In this paper, we determine the girth, chromatic number, clique number, perfect code, and perfectness of the Cayley sum graphs [ ] and [ ] (where [ ] denotes the set of all nonzero maximal ideals of [ ]). Finally, we classify the family of rings whose Cayley sum graphs are split, co-graph, threshold, chordal and claw-free.

The (strong) metric dimension of a connected graph [Formula: see text] is the cardinal number of a (strong) metric basis in [Formula: see text], i.e., the smallest subset of vertices which can determine the position of each vertex by its vector of distances with respect to itself in a unique way (the smallest subset of vertices such that for every pair of vertices [Formula: see text] contains a vertex [Formula: see text] such that either [Formula: see text] or [Formula: see text]. The essential annihilating-ideal graph [Formula: see text] of a commutative unital ring [Formula: see text] is a simple graph whose vertices are nonzero ideals of [Formula: see text] with nonzero annihilator and two distinct vertices [Formula: see text] are joined by an edge if and only if [Formula: see text] is an essential ideal of [Formula: see text]. In this paper, the metric and strong metric dimensions of [Formula: see text] are presented in some explicit formulae, for various classes of rings [Formula: see text].

Let ( M, f, g ) be an ( m, n )-hypermodule over an ( m, n )-hyperring ( R, h, k ). A proper subhypermodule N of M is called n -ary 2-absorbing subhypermodule if whenever g ( r ₁ ^{n− 1} , m ) ⊆ N for some r ₁ ^{n− 1} ∈ R and m ∈ M , then either g ( r ₁ ^{n −1} , M ) ⊆ N or g ( r _i , m , 1 _R ^{( n −2)} ) ⊆ N for some i ∈ {1, . . ., n − 1}. Various properties of n -ary 2-absorbing subhy-permodules are investigated. In particular, it is shown that if N is a subhypermodule of an ( m, n )-hypermodule ( M, f, g ) over an ( m, n )- hyperring ( R, h, k ), then N is n -ary 2-absorbing if and only if whenever g(I ₁ , I ₂ , 1 _R ^{( n −3)} , L ) ⊆ N for some hyperideals I ₁ , I ₂ of R and subhyper- module L of M , then either g ( I ₁ , I ₂ , 1 _R ⁽ⁿ⁻³⁾ , M ) ⊆ N or g ( I ₁ , 1 _R ^{( n −2)} , L ) ⊆ N or g ( I ₂ , 1 _R ^{( n −2)} , L ) ⊆ N . Also, n -ary 2-absorbing subhypermodules in multiplication ( m, n )-hypermodules are studied.

Let [ ] be a commutative ring with unity. The prime ideal sum graph of the ring [ ] is the simple undirected graph whose vertex set is the set of all nonzero proper ideals of [ ] and two vertices [ ] and [ ] are adjacent if and only if [ ] is a prime ideal of [ ]. In this paper, we obtain the strong metric dimension of the prime ideal sum graph for various classes of Artinian nonlocal commutative rings.